Students are presented with an engineering challenge that asks them to develop a material and model that can be used to test the properties of aortic valves without using real specimens. Developing material that is similar to human heart valves makes testing easier for biomedical engineers because they can test new devices or ideas on the model valve instead of real heart valves, which can be difficult to obtain for research. To meet the challenge, students are presented with a variety of background information, are asked to research the topic to learn more specific information pertaining to the challenge, and design and build a (prototype) product. After students test their products and make modifications as needed, they convey background and product information in the form of portfolios and presentations to the potential buyer.

This task provides a context for some of the questions asked in "6.NS Multiples and Common Multiples." A scaffolded version of this task could be adapted into a teaching task that could help motivate the need for the concept of a common multiple.

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

The purpose of this problem is to have students compose functions using tables of values only.

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

Students review information learned during the past five lessons and activities of the Introduction to Engineering unit. Working in teams, they create flyers and short quizzes about various types of engineering to share with the class and collect into a "Olympic Engineering Binder" for the class to keep.

Air is one of Earth's most precious resources, and we need to take care of it in order to preserve the environment and protect human health. To this end, students develop their understanding of visible air pollutants with an incomplete combustion demonstration, a "smog in a jar" demonstration, and by building simple particulate matter collectors.

In the first of two lessons of this curricular unit, students are introduced to the concept of friction as a force that impedes motion when two surfaces are in contact. Student teams use spring scales to drag objects, such as a ceramic coffee cup, along a table top or the floor, measuring the frictional force that exists between the moving object and the surface it slides on. By modifying the bottom surface of the object, students find out what kinds of surfaces generate more or less friction. They also discover that both static and kinetic friction are involved when an object initially at rest is caused to slide across a surface. In the second lesson of the unit, students design and conduct experiments to determine the effects of weight and surface area on friction. They discover that weight affects normal friction (the friction that results from surface roughness), but for very smooth surfaces, the friction due to molecular attraction is affected by contact area.

This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

Students take a closer look at cars and learn about some characteristics that affect their energy efficiency, including rolling resistance and the aerodynamics of shape and size. They come to see how vehicles are one example of a product in which engineers are making changes and improvements to gain greater efficiency and thus require less energy to operate.

Students model and design the sound environment for a room. They analyze the sound performance of different materials that represent wallpaper, thick curtains, and sound-absorbing panels. Then, referring to the results of their analysis, they design another room based on certain specifications, and test their designs.

This course explores the foundations of policy making in developing countries. The goal is to spell out various policy options and to quantify the trade-offs between them. We will study the different facets of human development: education, health, gender, the family, land relations, risk, informal and formal norms and institutions. This is an empirical class. For each topic, we will study several concrete examples chosen from around the world. While studying each of these topics, we will ask: What determines the decisions of poor households in developing countries? What constraints are they subject to? Is there a scope for policy (by government, international organizations, or non-governmental organizations (NGOs))? What policies have been tried out? Have they been successful?

There is a natural (and complicated!) predator-prey relationship between the fox and rabbit populations, since foxes thrive in the presence of rabbits, and rabbits thrive in the absence of foxes. However, this relationship, as shown in the given table of values, cannot possibly be used to present either population as a function of the other. This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations.

The example of rabbits and foxes was introduced in the task (8-F Foxes and Rabbits) to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation.

The example of rabbits and foxes was introduced in 8-F Foxes and Rabbits to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.

Urban governance comprises the various forces, institutions, and movements that guide economic and physical development, the distribution of resources, social interactions, and other aspects of daily life in urban areas. This course examines governance from legal, political, social, and economic perspectives. In addition, we will discuss how these structures constrain collective decision making about particular urban issues (immigration, education‰Ű_). Assignments will be nightly readings and a short paper relating an urban issue to the frameworks outlined in the class.

This series of 5 word problems lead up to the final problem. Most students should be able to answer the first two questions without too much difficulty. The decimal numbers may cause some students trouble, but if they make a drawing of the road that the girls are riding on, and their positions at the different times, it may help. The third question has a bit of a challenge in that students won't land on the exact meeting time by making a table with distance values every hour. The fourth question addresses a useful concept for problems involving objects moving at different speeds which may be new to sixth grade students.