In this task students must investigate this conjecture to discover that it does not work in all cases: Pick any two integers. Look at the sum of their squares, the difference of their squares, and twice the product of the two integers you chose. Those three numbers are the sides of a right triangle.

This Nrich game is designed to get children used to moving along a number line either side of a central point adding and subtracting within 27. It can help to introduce the idea of negative numbers or practice fluency with negatives by changing the midpoint to 0.

This Nrich problem introduces an intriguing trick which provides a context for practicing manipulation of fractions. Watching the video, or perhaps trying the trick out for themselves, can engage students' curiosity, and lead to some intriguing mathematics to explore and explain.

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4).

This Nrich problem is a fantastic opportunity for learners to apply knowledge of place value and offers a context for learning and practicing the relevant vocabulary (odd, even, multiple). The interactivity will help learners satisfy their curiosity in the sense of finding a 'better' solution, as it enables them to play around with the digits without having to commit anything to paper. This act of deciding whether one solution is better than another also provides a meaningful context in which to compare and order numbers.

This Nrich investigation is one that uses the very popular multilink cubes. It gives a wonderful opportunity for pupils to explore ways of recording. This activity is also designed to nurture children's curiosity by introducing mathematics into a familiar non-mathematical context. Children might end up pursuing different ideas from each other and this freedom to explore may well encourage learners to persevere more than they might usually. In this way, they will immerse themselves in the particular number sequence they have chosen to use, which will help them gain a deep understanding of its structure. This activity lends itself to pupils posing their own questions in the form “I wonder what would happen if...?”

Using manipulatives, students figure out patterns in problems involving seating arrangements. They write algebraic equations to describe the relationships.

Students are introduced to algebraic expressions that use more than one variable and have multiple solutions. They figure out combinations of two items at different costs, with each combination adding up to 100.

Prepared with pre-algebra or algebra 1 classes in mind, this module leads students through the process of graphing data and finding a line of best fit while exploring the characteristics of linear equations in algebraic and graphic formats. Then, these topics are connected to real-world experiences in which people use linear functions. During the module, students use these scientific concepts to solve the following hypothetical challenge: You are a new researcher in a lab, and your boss has just given you your first task to analyze a set of data. It being your first assignment, you ask an undergraduate student working in your lab to help you figure it out. She responds that you must determine what the data represents and then find an equation that models the data. You believe that you will be able to determine what the data represents on your own, but you ask for further help modeling the data. In response, she says she is not completely sure how to do it, but gives a list of equations that may fit the data. This module is built around the legacy cycle, a format that incorporates educational research feindings on how people best learn.

Investigate the effect of gravity on objects of various mass during free fall. Predict what the position-time and velocity-time graphs will look like. Compare graphs for light and heavy objects. Was Galileo Right is the last of five SmartGraphs activities designed for a typical physical science unit of study on the motion of objects.

The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout.

This Nrich activity offers free exploration which will help learners develop a deep understanding of halves and halving. The task gives a context in which to discuss the importance of the part-whole relationship of fractions so that children realise halves can be different sizes, depending on the whole.

At its most basic this Nrich task is an exercise in reading and recording information from a table. It also offers opportunities for children to do some elementary reasoning as they compare results with each other and work out why they differ.

The Nrich interactivity offers an ideal context in which to observe the "messy" randomness of results after a small number of experiments, and the predictability of results after a large number of trials. The problem also offers a good starting point for considering different probability distributions and their features, which could be followed up with the tasks Which List is Which and Data Matching.

This Nrich activity would be a good start to, or could follow on from, work on finding percentages of quantities. In whichever context it is used, it will provide a good assessment opportunity for you to find out what your pupils understand well and what they have difficulties with.

The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question.

This Nrich problem challenges learners to interpret data carefully and to search for all the information that there is implicit in any graphical representation. The question encourages them to contrast different ways of representing similar data and helps to make explicit their interpretation of what the data represents in order to solve the problem.

This task looks at zeroes and factorization of a general polynomial. It is related to a very deep theorem in mathematics, the Fundamental Theorem of Algebra, which says that a polynomial of degree d always has exactly d roots, provided complex numbers are allowed as roots and provided roots are counted with the proper "multiplicity.''

The intention of this task is to provide extra depth to the standard A-APR.2 it is principally designed for instructional purposes only. The students may use graphing technology: the focus, however, should be on what happens to the function g when x=0 and the calculator may or may not be of help here (depending on how sophisticated it is!).