This task provides a good entry point for students into representing quantities in contexts with variables and expressions and building equations that reflect the relationships presented in the context.

Statistics is the study of variability. Students who understand statistics need to be able to identify and pose questions that can be answered by data that vary. The purpose of this task is to provide questions related to a particular context (a jar of buttons) so that students can identify which are statistical questions. The task also provides students with an opportunity to write a statistical question that pertains to the context.

This is a task where it would be appropriate for students to use technology such as a graphing calculator or GeoGebra, making it a good candidate for students to engage in Standard for Mathematical Practice 5 Use appropriate tools strategically.

There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly.

The emphasis in this task is on the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter. It is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.

Comprehensive bibliographic full text reference and Canadian current events.

This task is intended for instructional (rather than assessment) purposes, providing an opportunity to discuss technology as it relates to irrational numbers and calculations in general. The task gives a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.

Students can sometimes have emotional outbursts in school settings. This fact will not surprise many teachers, who have had repeated experience in responding to serious classroom episodes of student agitation. Such outbursts can be attributed in part to the relatively high incidence of mental health issues among children and youth. It is estimated, for example, that at least one in five students in American schools will experience a mental health disorder by adolescence (U.S. Department of Health and Human Services, 1999). But even students not identified as having behavioral or emotional disorders may occasionally have episodes of agitation triggered by situational factors such as peer bullying, frustration over poor academic performance, stressful family relationships, or perceived mistreatment by educators.

This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. The goal of the task is to stimulate a conversation about rounding and about how to record numbers with an appropriate level of accuracy, tying in directly to the standard N-Q.3. It is therefore better suited for instruction than for assessment purposes.

After completing the associated lesson and its first associated activity, students are familiar with the 20 major bones in the human body knowing their locations and relative densities. When those bones break, lose their densities or are destroyed, we look to biomedical engineers to provide replacements. In this activity, student pairs are challenged to choose materials and create prototypes that could replace specific bones. They follow the steps of the engineering design process, researching, brainstorming, prototyping and testing to find bone replacement solutions. Specifically, they focus on identifying substances that when combined into a creative design might provide the same density (and thus strength and support) as their natural counterparts. After iterations to improve their designs, they present their bone alternative solutions to the rest of the class. They refer to the measured and calculated densities for fabricated human bones calculated in the previous activity, and conduct Internet research to learn the densities of given fabrication materials (or measure/calculate those densities if not found online).

Students construct three-dimensional models of water catchment basins using everyday objects to form hills, mountains, valleys and water sources. They experiment to see where rain travels and collects, and survey water pathways to see how they can be altered by natural and human activities. Students discuss how engineers design structures that impact water collection, as well as systems that clean and distribute water.

Students model a water catchment basin and survey water pathways to understand factors that affect water flow.

Students drop marbles into holes cut into shoebox lids and listen carefully to try to determine the materials inside the box that the marbles fall onto, illustrating the importance of surface composition on dolphins' abilities to sense materials, depth and texture using echolocation. This activity builds on what students learned in the associated lesson about bycatching by fisheries and how it affects marine habitats and species, especially dolphins. Students learn how echolocation works, why certain animals use it to determine the size, shape and distance of objects, and how people can take advantage of dolphins' echolocation ability when developing bycatch avoidance methods.

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task.

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.

The purpose of this task is to use finite geometric series to investigate an amazing mathematical object that might inspire students' curiosity. The Cantor Set is an example of a fractal.

In the exploration of ways to use solar energy, students investigate the thermal energy storage capacities of different test materials to determine which to use in passive solar building design.

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.