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.

In this lesson, students learn that navigational techniques change when people travel to different places land, sea, air and in space. For example, an explorer traveling by land uses different methods of navigation than a sailor or an astronaut.

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.

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.

Water is a limited resource that we use over and over again. The idea is to teach the science behind the water cycle, where water comes from and is located on the Earth. After research and developing and understanding of conservation students will create a water tower that will collect and store rainwater. Students will also create a Public Service Announcement (PSA) on water conservation.

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 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.

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.

This lesson introduces students to the concept of air pressure. Students will explore how air pressure creates force on an object. They will study the relationship between air pressure and the velocity of moving air.

Few people are aware of how crucial the sense of smell is to identifying foods, or the adaptive value of being able to identify a food as being familiar and therefore safe to eat. In this lesson and activity, students conduct an experiment to determine whether or not the sense of smell is important to being able to recognize foods by taste. The teacher leads a discussion that allows students to explore why it might be adaptive for humans and other animals to be able to identify nutritious versus noxious foods. This is followed by a demonstration in which a volunteer tastes and identifies a familiar food, and then attempts to taste and identify a different familiar food while holding his or her nose and closing his or her eyes. Then, the class develops a hypothesis and a means to obtain quantitative results for an experiment to determine whether students can identify foods when the sense of smell has been eliminated.

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.

As part of a (hypothetical) challenge to help a city find the most affordable and environmentally friendly way to clean up an oil spill, students design and conduct controlled experiments to quantify capillary action in sand. Like engineers and entrepreneurs, student teams use affordable materials to design and construct models to measure the rate of capillary action in four types of sand: coarse, medium, fine and mixed. After observing and learning from a teacher-conducted capillary tube demonstration, teams are given a selection of possible materials and a budget to work within as they design their own experimental setups. After the construction of their designs, they take measurements to quantify the rate of capillary action, create graphs to analyze the data, and make concluding recommendations. Groups compare data and discuss as a class the pros and cons of their designs. Pre- and post-evaluations and two worksheets are provided.