This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In the picture below, points $A$ and $B$ are the centers of two circles and they are collinear with point $C$. Also $D$ and $E$ lie on the two respecti...

In this open-ended, hands-on activity that provides practice in engineering data analysis, students are given gait signature metric (GSM) data for known people types (adults and children). Working in teams, they analyze the data and develop models that they believe represent the data. They test their models against similar, but unknown (to the students) data to see how accurate their models are in predicting adult vs. child human subjects given known GSM data. They manipulate and graph data in Excel® to conduct their analyses.

Student teams use sensors—motion detectors and accelerometers—to collect walking gait data from group members. They import their collected position and acceleration data into Excel® for graphing and analysis to discover the relationships between position, velocity and acceleration in the walking gaits. Then they apply their understanding of slopes of secant lines and Riemann sums to generate and graph additional data. These activities provide practice in the data collection and analysis of systems, similar to the work of real-world engineers.

While students need to be able to write sentences describing ratio relationships, they also need to see and use the appropriate symbolic notation for ratios. If this is used as a teaching problem, the teacher could ask for the sentences as shown, and then segue into teaching the notation. It is a good idea to ask students to write it both ways (as shown in the solution) at some point as well.

Presents the anatomy, physiology, biochemistry, biophysics, and bioengineering of the gastrointestinal tract and associated pancreatic, liver, and biliary systems. Emphasis on the molecular and pathophysiological basis of disease where known. Covers gross and microscopic pathology and clinical aspects. Formal lectures given by core faculty, with some guest lectures by local experts. Selected seminars conducted by students with supervision of faculty. Permission of instructor required. (Only HST students may register under HST.120, graded P/D/F.) The most recent knowledge of the anatomy, physiology, biochemistry, biophysics, and bioengineering of the gastrointestinal tract and the associated pancreatic, liver and biliary tract systems is presented and discussed. Gross and microscopic pathology and the clinical aspects of important gastroenterological diseases are then presented, with emphasis on integrating the molecular, cellular and pathophysiological aspects of the disease processes to their related symptoms and signs.

A gear is a simple machine that is very useful to increase the speed or torque of a wheel. In this activity, students learn about the trade-off between speed and torque when designing gear ratios. The activity setup includes a LEGO(TM) MINDSTORMS(TM) NXT pulley system with two independent gear sets and motors that spin two pulleys. Each pulley has weights attached by string. In a teacher demonstration, the effect of adding increasing amounts of weight to the pulley systems with different gear ratios is observed as the system's ability to lift the weights is tested. Then student teams are challenged to design a gear set that will lift a given load as quickly as possible. They test and refine their designs to find the ideal gear ratio, one that provides enough torque to lift the weight while still achieving the fastest speed possible.

The principles of genetics with application to the study of biological function at the level of molecules, cells, and multicellular organisms, including humans. Structure and function of genes, chromosomes and genomes. Biological variation resulting from recombination, mutation, and selection. Population genetics. Use of genetic methods to analyze protein function, gene regulation and inherited disease.

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal.

Students investigate motors and electromagnets as they construct their own simple electric motors using batteries, magnets, paper clips and wire.

Students are introduced to gear transmissions and gear ratios using LEGO MINDSTORMS(TM) NXT robots, gears and software. They discover how gears work and how they can be used to adjust a vehicle's power. Specifically, they learn how to build the transmission part of a vehicle by designing gear trains with different gear ratios. Students quickly recognize that some tasks require vehicle speed while others are more suited for vehicle power. They are introduced to torque, which is a twisting force, and to speed the two traits of all rotating engines, including mobile robots using gears, bicycles and automobiles. Once students learn the principles behind gear ratios, they are put to the test in two simple design activities that illustrate the mechanical advantages of gear ratios. The "robot race" is better suited for a quicker robot while the "robot push" calls for a more powerful robot. A worksheet and post-activity quiz verify that students understand the concepts, including the tradeoff between torque and speed.

Students take part in a hypothetical scenario that challenges them to inform customers at a local restaurant of how their use and disposal of plastics relates/contributes to the Great Pacific garbage patch (GPGP). What students ultimately do is research information on the plastics pollution in the oceans and present that information as a short, eye-catching newsletter suitable to hand out to restaurant customers. This activity focuses on teaching students to conduct their own research on a science-technology related topic and present it in a compelling manner that includes citing source information without plagiarism. By doing this, students gain experience and skills with general online searching as well as word processing and written and visual communication.

The first of these word problems is a multiplication problem involving equal-sized groups. The next two reflect the two related division problems, namely, "How many groups?" and "How many in each group?"

The purpose of this task is to show three problems that are set in the same kind of context, but the first is a straightforward multiplication problem while the other two are the corresponding "How many groups?" and "How many in each group?" division problems.

This hands-on activity explores five different forms of erosion (chemical, water, wind, glacier and temperature). Students rotate through stations and model each type of erosion on rocks, soils and minerals. The students record their observations and discuss the effects of erosion on the Earth's landscape. Students learn about how engineers are involved in the protection of landscapes and structures from erosion. Math problems are included to help students think about the effects of erosion in real-world scenarios.

This task gives students an opportunity to work with volumes of cylinders, spheres and cones. Notice that the insight required increases as you move across the three glasses, from a simple application of the formula for the volume of a cylinder, to a situation requiring decomposition of the volume into two pieces, to one where a height must be calculated using the Pythagorean theorem.

Many of our readers will no doubt already be familiar with MIT˘ď‹ď_s excellent OpenCourseWare (OCW), which offers free college-level curricula online to the public. The OpenCourseWare site is well worth a browse, as it offers courses on a variety of useful and engaging subjects such as business, health and medicine, mathematics, fine arts and science. This particular course, which was originally offered to undergraduate students in the spring of 2012, looks at the science behind global warming. Content includes lecture notes, assignments and student projects. The content could be used as a springboard for instructors teaching similar classes, or may prove useful to curious individuals looking to learn more about this timely and important topic.

Student teams learn about engineering design of green fluorescent proteins (GFPs) and their use in medical research, including stem cell research. They simulate the use of GFPs by adding fluorescent dye to water and letting a flower or plant to transport the dye throughout its structure. Students apply their knowledge of GFPs to engineering applications in the medical, environmental and space exploration fields. Due to the fluorescing nature of the dye, plant life of any color, light or dark, can be used unlike dyes that can only be seen in visible light.

This is a simple task addressing the distinction between correlation and causation. Students are given information indicating a correlation between two variables, and are asked to reason out whether or not a causation can be inferred. The task would be well-suited either as an introduction to this distinction, or as an assessment item.