Heat, cool and compress atoms and molecules and watch as they change between solid, liquid and gas phases.

The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.

This problem can be solved in more than one way. The choice in solution method may reflect the comfort and mathematical sophistication of the student.

This is a multi-step problem since it requires more than two steps no matter how it is solved. The problem is not scaffolded for the student, but each step is straightforward and should follow from the previous with a careful reading of the problem.

Explore stretching just a single strand of DNA using optical tweezers or fluid flow. Experiment with the forces involved and measure the relationship between the stretched DNA length and the force required to keep it stretched. Is DNA more like a rope or like a spring?

In this task students design a plan to conduct a random sample of the students in their school to estimate the proportion of students who think their parents are strict.

What happens when sugar and salt are added to water? Pour in sugar, shake in salt, and evaporate water to see the effects on concentration and conductivity. Zoom in to see how different sugar and salt compounds dissolve. Zoom in again to explore the role of water.

This task provides a familiar context allowing students to visualize multiplication of a fraction by a whole number. This task could form part of a very rich activity which includes studying soda can labels.

In this task students must figure out how much fresh water you must add to get a particular salt concentration in aquarium water.

This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180_.

Parts (d) and (e) of this task constitute a very advanced application of the skill of making use of structure: in (d) students are being asked to use the defining property of even and odd functions to manipulate expressions involving function notation. In (e) they are asked to see the structure in the system of two equations involving functions.

The intent of this problem is to have students think about how function addition works on a fundamental level, so formulas have been omitted on purpose.

The purpose of this task is to address the concept of opportunity cost through a real world context involving money.

This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.

The goal of this task is to help students understand the commutative property of addition by examining the addition facts for single digit numbers. This is important as it gives students a chance, at a young age, to do more than memorize these arithmetic facts which they will use throughout their education.

This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches. Teachers should of course feel free to provide additional scaffolding to encourage solutions or thinking in one particular direction. We include three solutions which fall into two general approaches, one based on reference to previously-derived results (e.g., the Pythagorean Theorem), and another conducted in terms of the geometry of rigid transformations.

The construction of the tangent line to a circle from a point outside of the circle requires knowledge of a couple of facts about circles and triangles. First, students must know, for part (a), that a triangle inscribed in a circle with one side a diameter is a right triangle. This material is presented in the tasks ''Right triangles inscribed in circles I.'' For part (b) students must know that the tangent line to a circle at a point is characterized by meeting the radius of the circle at that point in a right angle: more about this can be found in ''Tangent lines and the radius of a circle.''

This real world word problem about calculating a tip in a restaurant requires multiple steps.

This task is not about computing the final price of the shirt but about using the structure in the computation to make a general argument.

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.