Lesson Mathematical Modeling- Linear Approximations

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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NGSS: Next Generation Science Standards - Science

Common Core State Standards - Math

International Technology and Engineering Educators Association - Technology

State Standards

Worksheets and Attachments

Pre-Req 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 warm-up 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=xHtsuOB-TPw).

(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. Real-life 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 line-of-fit. For more advanced classes or for an extension, a conversation about finding the ‘best’ line-of-fit 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/least-squares-regression-line/ if more information is desired). For my purposes, this lesson is going to be implemented into a lower-level Algebra class and we will simply use judgement when drawing in an acceptable line-of-fit 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 x-axis (horizontal) and the dependent variable should go on the y-axis (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 line-of-fit, students use their two points of interest to first find the slope of their line-of-fit by doing the change in y-values divided by the change in x-values. Next, students use their slope, one point of interest, and the general slope-intercept form equation (y = mx + b) to solve for the b-value, or y-intercept. Now that the students have a slope and b-value, they can write their final equation that represents their line-of-fit. 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 line-of-fit, 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 pick-up 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).

line-of-fit: A line-of-fit (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

Pre-Lesson Assessment

Warm-up 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 warm-up 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

Wrapping-up Problem: Students will revisit the warm-up 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 warm-up. 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 warm-up to the wrapping-up 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 line-of-fit can be added to it)
• What is a line-of-fit 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 total-squared-error and apply the Least Square Method to find the best line of fit.

References

Copyright

Contributors

Supporting Program

Acknowledgements

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.