### Summary

The learning of linear functions is pervasive in most algebra classrooms. Linear functions are vital in laying the foundation for understanding the concept of modeling. This unit gives students the opportunity to make use of linear models in order to make predictions based on real-world data, and see how engineers address incredible and important design challenges through the use of linear modeling. Student groups act as engineering teams by conducting experiments to collect data and model the relationship between the wall thickness of the latex tubes and their corresponding strength under pressure (to the point of explosion). Students learn to graph variables with linear relationships and use collected data from their designed experiment to make important decisions regarding the feasibility of hydraulic systems in hybrid vehicles and the necessary tube size to make it viable.### Engineering Connection

Modeling is commonly used in mathematics and engineering and frequently used to make generalizations based on collected data. Currently, engineers of varied specializations are applying their expertise to create vehicles that require less fuel, making them more sustainable. Hydraulic hybrid vehicles use a hydraulic system with a hydraulic accumulator to store energy from the braking process and reuse it to accelerate the vehicle. In this unit, students learn about this engineering design and an engineering lab at a university that is researching the strength of latex used in such a hydraulic accumulator. The work includes designing and conducting experiments to collect data and model relationships between variables, such as the wall thickness of latex tubes and their corresponding strengths (pressure at the point of explosion), from which they can extrapolate the appropriate latex tube measurements required in order to store maximum energy in a modern hybrid passenger vehicle's braking system.

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Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 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 K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 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*.

###### Common Core State Standards - Math

- Interpret the parameters in a linear or exponential function in terms of a context. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

### Unit Overview

This "legacy cycle" unit is structured with a contextually based *grand challenge* followed by a sequence of instruction in which students first offer initial predictions (*generate ideas*) and then gather information from multiple sources (*multiple perspectives*). This is followed by *research and revise* as students integrate and extend their knowledge through a variety of learning activities. The cycle concludes with formative (*test your mettle*) and summative (*go public*) assessments that lead students towards answering the *challenge* question. See below for the progression of the legacy cycle through the unit. Research and ideas behind this way of learning may be found in *How People Learn: Brain, Mind, Experience and School*, (Bransford, Brown & Cocking, National Academy Press, 2000); see the entire text at http://www.nap.edu/catalog.php?record_id=9853.

The "legacy cycle" is similar to the "engineering design process" in that they both involve identifying existing societal needs or challenges, combining science and math to develop solutions, and using research conclusions to design optimal solutions. Though the engineering design process and the legacy cycle both result in viable solutions, they vary in how solution are devised and presented. See an overview of the engineering design process at https://www.nasa.gov/audience/foreducators/plantgrowth/reference/Eng_Design_5-12.html.

In the first lesson of this unit, students are introduced to the engineering challenge: To create small-scale models from which their testing results could be generalized to large-scale latex tubing for a hydraulic accumulator suitable for a passenger car. They learn about how hydraulic accumulators and hydraulic systems function, specifically how they conserve energy by capturing braking energy usually lost as heat. Through a nine-minute video, they watch and listen to an engineer talking about his lab-based model to test the feasibility of using an elastomer as an energy accumulator. Then they brainstorm and generate ideas for how to solve the challenge.

Through practice in the second lesson, students learn to quickly and efficiently interpret graphs and gain familiarity with common graph terminology such as independent variable, dependent variable, linear data, linear relationship, rate of change, as well as the equation for calculating slope. 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.

During the associated activity, student groups use latex tubes and bicycle pumps to conduct experiments to gather research data about the relationship between latex strength and air pressure. Students use this data to create graphs and linear models from which they extrapolate latex strength to answer the engineering design challenge question—to predict the latex dimensions that would be required for a full-size hydraulic accumulator installed in a modern passenger sedan. To conclude the unit, students describe the entire engineering analysis process that they experienced in order to answer the challenge, and explain their conclusions and how they could be applied to the design of hydraulic hybrid vehicles.

### Unit Schedule

Day 1: Latex and Hybrids: What's the Connection? lesson

Day 2: Variables and Graphs: What's Our Story? lesson

Days 3-4: Linear Models and Exploding Latex! activity

### Contributors

Erik Bowen, Carleigh Samson### Copyright

© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University### Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University### Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: January 23, 2018

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