This course provides a deep understanding of engineering systems at a level intended for research on complex engineering systems. It provides a review and extension of what is known about system architecture and complexity from a theoretical point of view while examining the origins of and recent developments in the field. The class considers how and where the theory has been applied, and uses key analytical methods proposed. Students examine the level of observational (qualitative and quantitative) understanding necessary for successful use of the theoretical framework for a specific engineering system. Case studies apply the theory and principles to engineering systems.

This final lesson in the unit culminates with the Go Public phase of the legacy cycle. In the associated activities, students use linear models to depict Hooke's law as well as Ohm's law. To conclude the lesson, students apply they have learned throughout the unit to answer the grand challenge question in a writing assignment.

In this simulation of a doctor's office, students play the roles of physician, nurse, patients, and time-keeper, with the objective to improve the patient waiting time. They collect and graph data as part of their analysis. This serves as a hands-on example of using engineering principles and engineering design approaches (such as models and simulations) to research, analyze, test and improve processes.

In this activity, kids will work on two fundamental early math skills – sorting/classifying, and graphing. There will also be some great fine motor skill practice! Includes place-based discussion questions, activity instructions, extension activities, songs, and student graph worksheets.

NGSS: K-LS1-1, 1-LS1-1, partially meets K-ESS3-1 (book and discussion)

Common Core: MP.4

Time: 45 minutes

Matierals: bag of dried beans ("16 bean soup"), paper bowls, glue, chart paper, the book "One Bean" or similar book about growing food plants, especially beans.

Student pairs are given 10 minutes to create the biggest box possible using one piece of construction paper. Teams use only scissors and tape to each construct a box and determine how much puffed rice it can hold. Then, to meet the challenge, they improve their designs to create bigger boxes. They plot the class data, comparing measured to calculated volumes for each box, seeing the mathematical relationship. They discuss how the concepts of volume and design iteration are important for engineers. Making 3-D shapes also supports the development of spatial visualization skills. This activity and its associated lesson and activity all employ volume and geometry to cultivate seeing patterns and understanding scale models, practices used in engineering design to analyze the effectiveness of proposed design solutions.

Students apply their knowledge of linear regression and design to solve a real-world challenge to create a better packing solution for shipping cell phones. They use different materials, such as cardboard, fabric, plastic, and rubber bands to create new “composite material” packaging containers. Teams each create four prototypes made of the same materials and constructed in the same way, with the only difference being their weights, so each one is fabricated with a different amount of material. They test the three heavier prototype packages by dropping them from different heights to see how well they protect a piece of glass inside (similar in size to iPhone 6). Then students use linear regression to predict from what height they can drop the fourth/final prototype of known mass without the “phone” breaking. Success is not breaking the glass but not underestimating the height by too much either, which means using math to accurately predict the optimum drop height.

This lab demonstrates Ohm's law as students set up simple circuits each composed of a battery, lamp and resistor. Students calculate the current flowing through the circuits they create by solving linear equations. After solving for the current, I, for each set resistance value, students plot the three points on a Cartesian plane and note the line that is formed. They also see the direct correlation between the amount of current flowing through the lamp and its brightness.

Students are introduced to the "Walk the Line" challenge question. They write journal responses to the question and brainstorm what information they need to answer the question. Ideas are shared with the class (or in pairs and then to the class, if class size is large). Then students read an interview with an engineer to gain a professional perspective on linear data sets and best-fit lines. Students brainstorm for additional ideas and add them to the list. With the teacher's guidance, students organize the ideas into logical categories of needed knowledge.

Students learn about complex networks and how to represent them using graphs. They also learn that graph theory is a useful mathematical tool for studying complex networks in diverse applications of science and engineering, such as neural networks in the brain, biochemical reaction networks in cells, communication networks, such as the internet, and social networks. Topics covered include set theory, defining a graph, as well as defining the degree of a node and the degree distribution of a graph.

A brief refresher on the Cartesian plane includes how points are written in (x, y) format and oriented to the axes, and which directions are positive and negative. Then students learn about what it means for a relation to be a function and how to determine domain and range of a set of data points.

Using a website simulation tool, students build on their understanding of random processes on networks to interact with the graph of a social network of individuals and simulate the spread of a disease. They decide which two individuals on the network are the best to vaccinate in an attempt to minimize the number of people infected and "curb the epidemic." Since the results are random, they run multiple simulations and compute the average number of infected individuals before analyzing the results and assessing the effectiveness of their vaccination strategies.

Students design and build model landfills using materials similar to those used by engineers for full-scale landfills. Their completed small-size landfills are "rained" on and subjected to other erosion processes. The goal is to create landfills that hold the most garbage, minimize the cost to build and keep trash and contaminated water inside the landfill to prevent it from causing environmental damage. Teams create designs within given budgets, test the landfills' performance, and graph and compare designs for capacity, cost and performance.

Student teams design and create LEGO® structures to house and protect temperature sensors. They leave their structures in undisturbed locations for a week, and regularly check and chart the temperatures. This activity engages students in the design and analysis aspects of engineering.

Students quantify the percent of light reflected from solutions containing varying concentrations of red dye using LEGO© MINDSTORMS© NXT bricks and light sensors. They begin by analyzing a set of standard solutions with known concentrations of food coloring, and plot data to graphically determine the relationship between percent reflected light and dye concentration. Then they identify dye concentrations for two unknown solution samples based on how much light they reflect. Students gain an understanding of light scattering applications and how to determine properties of unknown samples based on a set of standard samples.

Students use two different methods to determine the densities of a variety of materials and objects. The first method involves direct measurement of the volumes of objects that have simple geometric shapes. The second is the water displacement method, used to determine the volumes of irregularly shaped objects. After the densities are determined, students create x-y scatter graphs of mass versus volume, which reveal that objects with densities less than water (floaters) lie above the graph's diagonal (representing the density of water), and those with densities greater than water (sinkers) lie below the diagonal.

The CyberSquad tracks Digital position in time and then studies graphs to figure out what Hacker is scheming in this video from Cyberchase.

Students experiment with various ways to naturally dye materials using sources found in nature—roots, leaves, seeds, spices, etc.—as well as the method of extracting dyes. Then they analyze various materials using statistical methods and tackle an engineering design challenge—to find dyes that best suit the needs of a startup sustainable clothing company.

In this Cyberchase video segment, the CyberSquad figures out how to earn 100 snelfus in 11 days.

This course is about the mathematics that is most widely used in the mechanical engineering core subjects: An introduction to linear algebra and ordinary differential equations (ODEs), including general numerical approaches to solving systems of equations.