Algebra Links in Piezotronics

A TechMath Instructional Module

Adoracion C. Yumul, TeacherKeitha Whitaker, studentAnthony Pittman, student

John Driscoll, Plant ManagerPCB Piezotronics

Developed through support of the National Science Foundation, NSF

Logo, Grant Number

2007-2008

Preface

This TechMath Instructional Module was developed for use in a standard Algebra 1 class for high school students on a block schedule with 90 minutes of instructional time per lesson. The module was used as a culminating experience within a chapter on Analyzing Linear Equations. The content also supports the use of appropriate technology tools to efficiently collect, analyze, and display data and the use of English standard objectives in writing.

Applications came from PCB Piezotronics, a PCB group company located in eastern North Carolina. The company is a manufacturer of piezoelectric quartz sensors, accelerometers, and associated electronics for the measurement of dynamic pressure, force, and vibration.

The module has been piloted, revised, implemented, and revised by Adoracion C. Yumul and the TechMath team. We want to thank the National Science Foundation, TechMath team, ECU people, resource speakers, and students for their help during the piloting and revision.

Table of Contents

Lesson One: PCB Walk - A Fun day at PCB Piezotronics

Lesson Two: Let’s Get Real Into the World of

Business! - Graphing Linear Data

Lesson Three: Mr. Best Fit of the Future-

Scatter Plots and Lines of Fit

Lesson 1 – PCB Walk - A Fun day at PCB Piezotronics

Overview:

Students pick up so much information during the field trip. Planning an educationally sound field trip is obviously more work for the teacher, but reaps huge rewards for the students. With practice, the students will begin to formulate their own questions and construct their own meanings for every experience that comes their way. Algebra 1 students will take a tour at the PCB Piezotronics as an initial point for them to see the connections of Math in the workplace.

This lesson is optional for those who can not take the students on a field trip due to distance problem. Other option for teachers will be for them to let the students research the company on the Internet and come up with a brochure from those findings. PCB Piezotronics Company is located at 10869 Highway 903 Halifax, NC.

NCSCOS: Social Studies Goal 9 - The learner will analyze factors influencing the United States economy.Objective 9.01

Identify phases of the business cycle and the economic indicators used to measure economic activities and trends.

Social Studies Goal 5 – The learner will acquire strategies needed for effective incorporation of computer technology in the learning process.Objective 5.01

Use word processing to create, format, and produce classroom assignments/projects.

Learning Objectives:

♦ To broaden and deepen the mathematical background of students through PCB Piezotronics.

♦ To prepare students to work in fields where specific and broadly based advanced mathematical knowledge is required.

♦ To enable the students to appreciate the coherence, logical structure, and broad applicability of mathematics in the workplace.

Materials/Resources:

Paper sheets, ink pen, camera, journals, handout

Assignment/Activity:

During the Trip:

Look for answers to investigation questions by using the KWL Chart.

Name: _______________________________________ Date: ____________________

KWL Chart

Teacher activates students' prior knowledge by asking them what they already Know; then students (collaborating as a classroom unit or within small groups) set goals specifying what they Want to learn; and after the trip, discuss what they have Learned. Students apply higher-order thinking strategies that help them construct meaning from what they have seen and observed from the trip and help them monitor their progress toward their goals. A worksheet is given to every student that includes columns for each of these activities.

KWhat I KNOW

WWhat I WANT to Know

LWhat I LEARNED

     

K W L__ 1.

__ 2.

__ 3.

__ 4.

__ 5.

__ 1.

__ 2.

__ 3.

__ 4.

__ 5.

 

After the trip, go back to the "K" column and see if any of your prior knowledge was inaccurate. Check any of them that are inaccurate, based on the information you got from the trip. Rewrite any of your statements that were inaccurate so they are correct.Then go to the "W"

column and check any of your questions that the field trip did not answer. Be prepared to bring these unanswered questions up in class, or tell how you will find answers to them and where you will look to get the answers.

Use a focused journal. Help students interact with their adventure using all their senses.

*Talking*Drawing*Writing phrases or quick observations*Taking pictures(Note: The teacher should check with the company’s policy if taking pictures is

allowed.)

After the trip: Use data collected to create a meaningful brochure. Plan a brief presentation of the brochure to an audience. Rubric for the Brochure

Assessment:Students’ output which is the brochure making will be graded based on the rubric.

They will also send an email to the teacher about their meaningful experience during the trip. The post-trip journal will be graded based on the very last part of the rubric.

CATEGORY 4 3 2 1

Writing - Organization

Each section in the brochure has a clear beginning, middle, and end.

Almost all sections of the brochure have a clear beginning, middle and end.

Most sections of the brochure have a clear beginning, middle and end.

Less than half of the sections of the brochure have a clear beginning, middle and end.

Writing - Grammar There are no grammatical mistakes in the brochure.

There are no grammatical mistakes in the brochure after feedback from an adult.

There are 1-2 grammatical mistakes in the brochure even after feedback from an adult.

There are several grammatical mistakes in the brochure even after feedback from an adult.

Spelling & Proofreading

No spelling errors remain after one person other than the typist reads and corrects the brochure.

No more than 1 spelling error remains after one person other than the typist reads and corrects the brochure.

No more than 3 spelling errors remain after one person other than the typist reads and corrects the brochure.

Several spelling errors in the brochure.

Writing - Vocabulary The authors correctly use several new mathematical words and define these words unfamiliar to the reader.

The authors correctly use a few new mathematical words and define these words unfamiliar to the reader.

The authors try to use some new mathematical vocabulary, but may use 1-2 words incorrectly.

The authors do not incorporate new mathematical vocabulary.

Content - Accuracy All facts in the brochure are accurate and show mathematical ideas.

99-90% of the facts in the brochure are accurate and show less mathematical ideas.

89-80% of the facts in the brochure are accurate and barely show mathematical ideas.

Fewer than 80% of the facts in the brochure are accurate and do not show mathematical ideas.

Attractiveness & Organization

The brochure has exceptionally attractive formatting and well-organized information.

The brochure has attractive formatting and well-organized information.

The brochure has well-organized information.

The brochure's formatting and organization of material are confusing to the reader.

Post-trip Journal Uses complex and refined mathematical ideas and reasoning. Explanation is detailed and clear.

Uses effective mathematical ideas and reasoning. Explanation is clear.

Some evidence of mathematical ideas and reasoning. Explanation is a little difficult to understand, but includes critical

Little evidence of mathematical reasoning. Explanation is difficult to understand and is missing several components OR was

Making A Brochure : HCSE Field Trip Tour @ PCBTeacher Name: _________________________________________ Student Name:     _________________________________________

components. not included.

Lesson 2 – Let’s Get Real Into the World of Business! - Graphing Linear Data

Overview:In this topic, students will employ factual information records accumulated from

PCB Piezotronics to create linear graphical illustrations. The students will utilize TI-84 Graphing Calculator and an interactive applet found at Shodor’s website which is called Graph Sketcher. This lesson is intended to teach students the concept of functions and their illustrations as rules and data tables, as well as the mathematical ideas of independent and dependent variables.

NCSCOS: Algebra I Goal 4 - The learner will use relations and functions to solve problems.

Objective 4.01 - Use linear functions or inequalities to model and solve problems; justify results.

Solve using tables, graphs, and algebraic properties. National Council of Teachers of Mathematics Standards:

♦ Understand patterns, relationships and functions.♦ Represent and analyze mathematical situations and structures using algebraic symbols.♦ Use mathematical models to represent and understand quantitative relationships.♦ Analyze change in various contexts.

Learning Objectives:

Students will interpret graphs of functions and draw graphs of functions based on the data taken from PCB Piezotronics.

Materials/Resources:

TI-84 Graphing Calculator, computer, Internet access, student data sheet (PCB), graphing worksheets, colored pencils, transparencies, flip chart

Key Terms:

This lesson introduces students to the following terms through the included discussions:

x-intercept The x-coordinate of the point where the line crosses the x-axis

y-intercept

The y-coordinate of the point where the line crosses the y-axis

linear function A function of the form f(x) = mx + b where m and b are some fixed numbers. The names "m" and "b" are traditional. Functions of this kind are called "linear" because their graphs are straight lines

constantA term or expression with no variables. In math, things that do not change: for

example distance, volume, mass, are called constants.

correlationA statistical measure referring to the relationship between two random variables.

It is a positive correlation when each variable tends to increase or decrease as the other does, and a negative or inverse correlation if one tends to increase as the other decreases.

independent variableA variable in an equation that may have its value freely chosen without

considering values of any other variable. For equations such as y = 3x – 2, the independent variable is x.

dependent variableA variable that depends on one or more other variables. For equations such as

y = 3x – 2, the dependent variable is y. The value of y depends on the value chosen for x. Usually the dependent variable is isolated on one side of an equation

slope of the linear function The slope of the line y = mx + b is the rate at which y is changing per unit of change in x. The units of measurement of the slope are units of y per unit of x

output The number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output

Instructional Plan:

Students will see a PowerPoint presentation about PCB Piezotronics regarding how the company runs, how does the management team use math for wage scales, in determining overtime versus new hire, forecasting, and in preparing reports. They will be given a sheet of paper that contains Student Notes for PowerPoint. As they view the presentation, they are to answer the guided questions on their student notes. Students will be introduced to PCB graphs that they will interpret and analyze and to some PCB data to draw on graphs that they will display on graphing calculators, manipulate on the computer, and finally to project on the LCD for group reporting.

Name: _______________________________ Block: ___________ Date: ______

STUDENT NOTES FOR PCB POWERPOINT PRESENTATIONAnswer the following questions.

1. What does PCB do? ________________________________________________________________________________________________________________________________________________

2. How PCB got its name? ________________________________________________________________________________________________________________________________________________

3. Define: pico _____________________________________________________________

coulomb ___________________________________________________________

4. Who can work at PCB? Describe what kind of tasks they perform. _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

5. Identify specific skills in math that are needed in order to get an employment at PCB._______________________________________________________________________________________________________________________________________________

Journal Topic“Describe specific connections between math and the world of

business.”

The following graph is a sample taken from PCB Piezotronics. Students will answer initially the question “how can real world business situations be modeled using graphs and functions?”

Questions for Students:

This graph fitted with the line is provided by PCB.

What does the graph represent? Analyze whether the line is a good fit for the data or not. Explain

your answer. Does the graph represent a linear function? Why/Why not? Find the week that has the highest number of overtime hours and

the week that has the least number of overtime hours. Estimate how many hours of overtime work are spent in 2007.

Explain how you find your possible estimate. Describe what is happening in the graph. Does it show positive

correlation, negative correlation, or no correlation at all?

Assessment:

1. In 2000, PCB Piezotronics shipped a total of 6515 of its standard product model. The shipping is increased by about 1000 each year until 2006.

a. Write an equation to find the total number of standard product model S shipped for any year t between 2000 and 2006.

b. If the trend continues, what will be the total standard model shipped in 2012?c. Use the TI-84 graphing calculator to describe your answer.

2. The table shows the 2007 monthly efficiency production of the workers at PCB from January to July. Their goal is to achieve 95% to 105% efficiency each week.

Month January February March April May June JulyPercent 140% 119% 137% 100% 105% 93% 120%

a. Make a scatter plot of the data.b. Find an appropriate linear function of the line and state the independent and

dependent variables.c. Use the linear function to predict the efficiency production for the month of

November.d. Do you believe that your linear function is appropriate for a month beyond the

range of data such as February 2008? Explain.

Lesson 3 – Mr. Best Fit of the Future- Scatter Plots and Lines of Fit

Overview:

At this point, students now understand how to plot points on a Cartesian plane. They also understand how to find the data trend on a Cartesian plane. These skills are essential to accomplish this lesson, determining the line of best fit. Again, data from PCB will be utilized for students to work on the relationship that exists in the data and to make predictions.

NCSCOS: Algebra I Goal 3 - The learner will collect, organize, and interpret data with matrices and linear models to solve problems.

Goal 4 - The learner will use relations and functions to solve problems.

Objective 4.01 Use linear functions or inequalities to model and solve problems; justify results.a. Solve using tables, graphs, and algebraic properties. b. Interpret constants and coefficients in the context of the problem.

National Council of Teachers of Mathematics Standards:

Algebra, grade 6-8

♦ Understand patterns, relations, and functions

Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations

Relate and compare different forms of representation for a relationship

♦ Use mathematical models to represent and understand quantitative relationships

Model and solve contextualized problems using various representations, such as graphs, tables and equations

Learning Objectives:

Students will interpret points on a scatter plot and write equations for lines of fit.

Materials/Resources:

TI-84 Graphing Calculator, computer, Internet access, Microsoft Excel, graphing worksheets, colored pens, ruler

Key Terms:

This lesson introduces students to the following terms through the included discussions:

Interactivate lessons from shodor website has many good discussions on these topics. See the following discussions linked to from this page: http://www.shodor.org/interactivate/discussions/

“Line of Best Fit”

“Bivariate Data Relations”

“Correlation Coefficients”

“Graphing and Interpreting Bivariate Data”

“Univariate Data and Bivariate Data”

“Using Residuals to Identify a Line of Best Fit”

scatter plot

A graph of ordered pairs involving two sets of data. It is used to investigate a relationship between two quantities.

positive correlation

In a scatter plot, as x increases, y increases.

negative correlation

In a scatter plot, as x increases, y decreases.

line of fit

A line that describes the trend of the data in a scatter plot.

Instructional plan:

Students will be given examples of scatter plots that show positive correlation, negative correlation, and no correlation. Then, a discussion will follow explaining how to find the line of best fit. They will examine more scatter plots, practice drawing a scatter plot, and determine what relationship exists, if any, in the data.

There are several ways to find the best-fitting line for a given set of data points. In this lesson, students will use a graphical approach.

Students will be introduced to Microsoft Excel to work on the data given to them.

Questions for Students:

1. How do scatter plots help identify trends in data?2. What would you do to find the equation of the line?3. What does the phrase linear relationship mean to you?4. How can you remember whether a set of data points shows a positive correlation

or a negative correlation?

Assessment:

A. The table shows the PCB Piezotronics Total Models shipped in 2007. At the outset, students will focus on the question, “Can we predict the level of productivity at PCB?” (I deleted the former data and used a new set of data instead.)

Month Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.Number 1010 820 1100 1050 1900 1900 1140 1350 1990 1420 1250 1240

1. Using Microsoft Excel, draw a scatter plot for NC Total Models shipped in 2007 and determine what relationship exists, if any.

2. Draw a line of fit for the scatter plot.3. Determine the slope-intercept form of an equation for the line of fit.4. Can you possibly predict the total models to be shipped for 2008 by using the

given data? Explain your answer.B. Pick two of the following graphs and analyze them. In an essay, write the

answer to the following question: “Is it appropriate to model the data with a line of best fit? Why / why not? Your essay will be graded based on the rubric below.

C. Mini-Project: What do you believe is the best way to illustrate the information in the graphs you chose? Re-express the data in a different type of graph and explain why you chose that type of graph. Be creative in presenting your graphs.

HALIFAX COUNTY SCHOOL OF ECOLOGYRubric For Assessing Journal Responses

Name: ________________________ Teacher: A. YUMULDate Submitted: ____________ Title of Work: ___________________ 

  Criteria Points

4 3 2 1  

ExplanationA complete response

with a detailed explanation.

Good solid response with

clear explanation.

Explanation is unclear.

Misses key points. ____

Use Of Visuals

Clear diagram or sketch with some

detail.

Clear diagram or sketch.

Inappropriate or unclear diagram.

No diagram or sketch. ____

Mechanics No math errors.No major math errors or serious

flaws in reasoning.

May be some serious math

errors or flaws in reasoning.

Major math errors or serious

flaws in reasoning.

____

Demonstrated Knowledge

Shows complete understanding of the

questions, mathematical ideas,

and processes.

Shows substantial understanding of

the problem, ideas, and processes.

Response shows some

understanding of the problem.

Response shows a complete lack of understanding for the problem.

____

RequirementsGoes beyond the

requirements of the problem.

Meets the requirements of

the problem.

Hardly meets the requirements of

the problem.

Does not meet the requirements of the problem.

____

Counter Examples

Includes counter examples. a

Does not include counter

examples.yy ____

        Total----> ____

 

Teacher Comments: