Quick Look
Grade Level: 9 (810)
Time Required: 1 hour
Approximately five minutes to complete a warmup, 45 minutes to engage in the lesson and 810 minutes to complete a wrappingup formative assessment.
Lesson Dependency: None
Subject Areas: Algebra, Number and Operations
Summary
Students investigate the idea of linear approximation. Students apply mathematical modeling, specifically linear approximation, to an already collected data set to make a prediction. In this lesson, students first engage in a warmup that is not a perfectly linear data set, being exposed to this for the first time. Students take on the role as a packaging engineer to learn the process to apply linear approximation modeling: collecting data, creating a graph, drawing a lineoffit, creating a model in the form of an equation, defining the model’s variables, and evaluating with the model. Students ultimately use their linear model to predict the net weight of cereal in grams contained by 260 square inches of cardboard packaging.Engineering Connection
Engineers deal with realworld problems, and often in realworld problems, numbers and data do not follow a perfect model like they often do in a mathematics classroom. To accommodate for this, engineers use mathematical modeling to investigate the relationship between variables, which allows them to make an accurate prediction of situational outcomes. Engineers follow the engineering design process while they work, and throughout this lesson, the process of collecting data, creating a model, testing the model, and making necessary amendments to the model posttesting are applicable to the engineering design process. In this lesson, students more specifically explore how packaging engineers apply mathematical modeling to help determine packaging variations for cereal and even offer a suggestion of linear approximation model to Battle Creek Cereal’s executive team that can be used to create skews of packaging based on square inches of cardboard packaging used and the net weight of the cereal in grams.
Learning Objectives
After this lesson, students should be able to:
 Investigate relationships between quantities by using points on scatter plots.
 Model an approximately linear situation.
 Apply lines of fit to make and evaluate predictions.
Educational Standards
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technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
NGSS: Next Generation Science Standards  Science

Communicate technical information or ideas (e.g. about phenomena and/or the process of development and the design and performance of a proposed process or system) in multiple formats (including orally, graphically, textually, and mathematically).
(Grades 9  12)
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Use mathematical representations of phenomena to describe explanations.
(Grades 9  12)
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Common Core State Standards  Math

Model with mathematics.
(Grades
K 
12)
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Create equations that describe numbers or relationships
(Grades
9 
12)
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Build a function that models a relationship between two quantities
(Grades
9 
12)
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Write a function that describes a relationship between two quantities
(Grades
9 
12)
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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
(Grades
9 
12)
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International Technology and Engineering Educators Association  Technology

Students will develop an understanding of the characteristics and scope of technology.
(Grades
K 
12)
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State Standards
Michigan  Math

Model with mathematics.
(Grades
K 
12)
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Create equations that describe numbers or relationships
(Grades
9 
12)
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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
(Grades
9 
12)
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Build a function that models a relationship between two quantities
(Grades
9 
12)
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Write a function that describes a relationship between two quantities
(Grades
9 
12)
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Do you agree with this alignment?
Worksheets and Attachments
Visit [www.teachengineering.org/lessons/view/mis2348mathematicalmodelinglinearapproximationslesson] to print or download.PreReq Knowledge
An understanding of linear functions  graphing, writing equations, and slope. An understanding of representing data as a table and scatter plot. An understanding of how to use a ruler.
Introduction/Motivation
(This should take place after the students have had time to work on their warmup problem and share out ideas). Now that you all have had a chance to hear some ideas, I want to start our lesson today with this video (play What is Math Modeling? Video Series Trailer https://www.youtube.com/watch?v=_1BFNNvg2ec).
The world around us is filled with important and unanswered questions. Mathematical modeling provides qualitative and quantitative understanding of these problems and can even predict future behavior. Who has ever had to make a prediction about something? How did you make your prediction? (educated guess, past experience, copied other predictions.) Making a guess at a family reunion for how much candy is in a jar might be fine, but in most occasions, a guess is not precise enough. Before we start today, I want you to get a quick insight into what math modeling is (play What is Math Modeling? Video Series Part 1: What is Math Modeling?,  https://www.youtube.com/watch?v=xHtsuOBTPw).
(Have students follow along on their Mathematical Modeling Linear Approximations Handout). Today you will investigate the idea of linear approximations. Often times in math class, the situations from a textbook follow a trend perfectly. Reallife mathematics often do not do this and require users to apply models that ‘fit’ a data set in order to make predictions about future data. This mathematical modeling is often used in engineering, such as by packaging engineers.
Your Task: Battle Creek Cereal has a variety of packaging sizes for their Crispy Puffs cereal. You have a list of six current packages. Though they like their current packaging sizes, they want to expand their options. They need your help to create a model that they can use to create more packaging options.
Lesson Background and Concepts for Teachers
For this lesson, the students will mathematically model a situation using a lineoffit. For more advanced classes or for an extension, a conversation about finding the ‘best’ lineoffit can be had. My method of choice to minimize the variance between the data points and regression line is the Least Squares Method (See http://www.statisticshowto.com/leastsquaresregressionline/ if more information is desired). For my purposes, this lesson is going to be implemented into a lowerlevel Algebra class and we will simply use judgement when drawing in an acceptable lineoffit for our data.
For the first step in the process, data collection, make sure students are aware of the variables in which a relationship exists. Students may ask how the variables should be arranged in the table, and this comes down to which variable is the independent variable and which variable is the dependent variable. In general, independent variables are the values that can be changed or controlled in a given model or equation. They provide the input, which is modified by the model to change the output. Likewise, dependent variables are the values that result from the independent variables. For graphing the scatter plot, the independent variable should go on the xaxis (horizontal) and the dependent variable should go on the yaxis (vertical). The scaling of the axes are important in order to get a nice visual, so importance should be placed on this aspect. For creating a model from the drawn lineoffit, students use their two points of interest to first find the slope of their lineoffit by doing the change in yvalues divided by the change in xvalues. Next, students use their slope, one point of interest, and the general slopeintercept form equation (y = mx + b) to solve for the bvalue, or yintercept. Now that the students have a slope and bvalue, they can write their final equation that represents their lineoffit. For the last step of evaluating, students are given a random input value and should apply the order of operations [PE(MD)(AS)] to evaluate and come to one final output value prediction.
Associated Activities
 Judgement with Jellybeans  Students collect data and use linear approximation to create a model. Students use their linear model to predict the number of Jelly Belly jelly beans that are in a similar cylindrical tube with a given height.
Lesson Closure
After today’s lesson, I want you all to think in your head and tell yourself how well you understand the following learning objectives for today’s lesson:
 Investigate relationships between quantities by using points on scatter plots.
 Model an approximately linear situation.
 Apply lines of fit to make and evaluate predictions.
Our lesson plays a key role in mathematics outside of the classroom and the idea of mathematical modeling will help you make a prediction in our next activity, called Judgement with Jelly Beans, where you all will take on the role of being a packaging engineer, and even have the chance to be paid! To be successful, you will need to apply the linear approximation process. Can somebody recap the steps of that process that we used today? (Answer: data collection, graph a scatter plot, draw a lineoffit, create a model, define the variables of the model, evaluate with the model to make a prediction.) As you leave class, try to think of other situations where it might be helpful to create a model in order to make a prediction and we will pickup there for the activity.
Vocabulary/Definitions
evaluate: To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations.
linear approximation: Linear approximation attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable (independent variable), and the other is considered to be a scalar response (dependent variable).
lineoffit: A lineoffit (or "trend" line) is a straight line that best represents the data on a scatter plot.
manufacturer: A person or company that makes goods for sale.
Assessment
PreLesson Assessment
Warmup Problem: In the Mathematical Modeling Linear Approximations Handout, students are introduced to a scenario of labor productivity that logistics engineers deal with. This may be the first time students work with data that is not a perfectly linear situation. Allow five minutes for students to work individually or with a partner on the warmup questions. Make sure students understand that there are no wrong answers and that this is merely to get their gears turning about how to deal with an approximately linear situation. Also make sure students pay attention to the table headings of hours worked per week and value of goods purchased in US dollars. Spend a couple minutes to allow students to verbally share their answers with the rest of the class so they can hear others’ ideas.
Lesson Summary Assessment
Wrappingup Problem: Students will revisit the warmup problem. This will allow students to now apply the learned linear approximation process to an approximately linear situation that they already have seen and thought about in the beginning of the class. Allow students eight to ten minutes to work on this individually or with the same partner that they worked with during the warmup. Allow students some time to verbally share their answers and allow some students to display their work on a document camera. Ask students how their thought process and ideas changed from the warmup to the wrappingup problem. Some potential discussion questions:
 Why is it important to graph the data? (To get a visual of the trend of the data and so a lineoffit can be added to it)
 What is a lineoffit and why is it important? (It is a line that best represents the trend of the data and it is important so that predictions can be made about data that is not represented or future data)
 Why do we need to define our variables in our model? (If you do not define your variables, the model will not have any meaning other than it being an equation of variables, numbers, and symbols. It is important to know what each variable represents so you know where values go when making a prediction)
Lesson Extension Activities
If students are really understanding the concepts, pose the question “Is there a ‘best’ line of fit to model approximately linear situations? If so, how do you find it?” For higher grades or more advanced algebra classes, students could learn how to calculate totalsquarederror and apply the Least Square Method to find the best line of fit.
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Students learn how to quickly and efficiently interpret graphs, which are used for everyday purposes as well as engineering analysis. The focus is on students becoming able to clearly describe linear relationships by using the language of slope and the rate of change between variables.
Students collect data and apply mathematical modeling, specifically linear approximation, to predict what will happen in a specific situation. In this activity, students collect data to determine the number of Jelly Belly jelly beans it takes to fill up the respective tube.
References
SIAM Connects. What Is Math Modeling? Video Series Part 1: What Is Math Modeling? Last modified July 8, 2016. YouTube. Accessed August 1, 2018. https://www.youtube.com/watch?v=xHtsuOBTPw.
SIAM Connects. What is Math Modeling? Video Series Trailer. Last modified July 8, 2016. YouTube. Accessed August 1, 2018. https://www.youtube.com/watch?time_continue=106&v=_1BFNNvg2ec.
Statistics How To. Least Squares Regression Line: Ordinary and Partial. Last modified June 25, 2018. Accessed August 1, 2018. http://www.statisticshowto.com/leastsquaresregressionline/.
Copyright
© 2020 by Regents of the University of Colorado; original © 2019 Michigan State UniversityContributors
William Harnica; Leyf StarlingSupporting Program
RET Program, College of Engineering, Michigan State UniversityAcknowledgements
This lesson was created due to funding from the National Science Foundation and Research Experience for Teachers program at Michigan State University in East Lansing, Michigan. NSF Site Research Experience for Teachers, College of Engineering, Michigan State University RET Grant#1609339.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Last modified: August 6, 2021
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