The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.

For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.

This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection. There are many other ways in which to exhibit the congruency and students and teachers are encouraged to explore the different possibilities.

This exercise demonstrates that judgment (non-random) samples tend to be biased in the sense that they produce samples that are not balanced with respect to the population characteristics of interest.

The purpose of this task is for students to understand the reason it makes sense for the product of two negative numbers to be positive. The idea is that if the properties of operations with which we are familiar when we do arithmetic with positive numbers are universal, then we have to define multiplication on signed numbers the way we do. The task only works through a single example, but the argument would work for any two negative numbers.

The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of raandom assingment to experimental groups in an experiment.

This problem allows students to see words that can describe the expression. Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction.

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.

The instructions for the two expressions sound very similar, however, the order in which the different operations are performed and the exact wording make a big difference in the final expression. Students have to pay close attention to the wording: Ňsubtract the result from 1Ó and Ňsubtract 1 from the resultÓ are very different.

The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example interpreting f(x) as the product of f and x.

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

This task is the first in a series that leads students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression in order to justify the steps in a solution (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding).

In this particular task, we are trying to get students to prove the zero product property, which is the lynchpin in understanding how to solve quadratic equations by factoring. In tasks that follow in this series, students will apply this property to solving quadratic equations and justifying their solutions.

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

This task is the fourth in a series of tasks that leads students to understand The Zero Product Property (ZPP) and apply it to solving quadratic equations. The emphasis is on using the structure of a factorable expression to justify the solution method (rather than memorizing steps without understanding). Teachers should feel free to skip any tasks in the series that students have already mastered.

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!).

For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x_r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact.

This task continues ``Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in ``Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions.