The purpose of this task is for students to practice setting up and solving right triangles for unknown sides and unknown angles. Students will use the Pythagorean Identity as well as trigonometric ratios to set up and solve equations.

The purpose of this task is for students to learn about piecewise functions using their background knowledge of domain, linear functions and function notation. Students will develop an understanding of piecewise-defined functions by

• Using their knowledge of domain to talk about the four ‘pieces’ of the graph that are made up of different linear functions.
• Creating a story for the graph, recognizing how and where the story changes based on each section of the graph.
• Interpreting values for different sections of the graph and identifying which equation from the piecewise function to use based on input/output values.

The purpose of this task is to surface ideas and representations for quadratic functions. The task is designed to elicit tables, graphs, and equations, both recursive and explicit to describe a growing pattern. The classroom discussion will focus on the growth shown in the various representations, developing the idea that quadratic functions show linear rates of change

The purpose of this task is to extend students’ understanding of the procedure of completing the square using area models by introducing situations in which the coefficient of the !! term is not 1. In the task, students will use area models to represent expressions in the form !!! + !" + ! and to generalize and apply the process of completing the square to several examples.

The purpose of this task is for students to practice determining whether one event is independent of another event. Students will use data from different representations, plus make sense of whether or not one scenario would be independent of another. In the end, students will explain how to quickly determine independence from a Venn diagram, a tree diagram, and a two-way table.

In this task, students are investigating what sums of different types of numbers will produce. The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.

In this learning cycle, students focus on classes of geometric figures that can be carried onto themselves by a transformation—figures that possess a line of symmetry or rotational symmetry. In this task the idea of “symmetry” is surfaced relative to finding lines that reflect a figure onto itself, or determining if a figure has rotational symmetry by finding a center of rotation about which a figure can be rotated onto itself. This work is intended to be experimental (e.g., folding paper, using transparencies, using technology, measuring with ruler and protractor, etc.), with the definitions of reflection and rotation being called upon to support students’ claims that a figure possesses some type of symmetry. The particular classes of geometric figures considered in this task are various types of quadrilaterals.

In this task, students continue to focus on classes of geometric figures that can be carried onto themselves by a transformation—figures that possess a line of symmetry or rotational symmetry. Students solidify the idea of “symmetry” relative to finding lines that reflect a figure onto itself, or determining if a figure has rotational symmetry by finding a center of rotation about which a figure can be rotated onto itself. They also look for and describe the structure that determines if a figure possesses some type of symmetry. This work can be experimental (e.g., folding paper, using transparencies, using technology, measuring with ruler and protractor, etc.), or theoretical, with the definitions of reflection and rotation being called upon to support students’ claims that a figure possesses some type of symmetry.

The particular classes of geometric figures considered in this task are various types of regular polygons, and students will look for patterns in the types and numbers of lines of symmetry a regular polygon with an odd number of sides possesses, versus those with an even number of sides. They should also note a pattern between the smallest angle of rotation that carries a regular polygon onto itself and the number of sides of the polygon.

Students will synthesize what they have learned about functions to select the correct function type in a series of modeling problems. Students must also draw on their study of statistics, using graphs and functions to model a context presented with data and/or tables of values. In this module, the modeling cycle is used as the organizing structure, rather than function type.

This online tutorial provides instruction and practice items on the side-splitter theorum, angle bisector theorum, and area and similarity.

The purpose of this task is to develop the quadratic formula as a way of finding xintercepts of a quadratic function that crosses the x-axis. In a future task this same quadratic formula will be used to find the roots of any quadratic, including those with complex roots whose graphs do not cross the x-axis. In this task, the quadratic formula is developed from the perspective of visualizing the distance the x-intercepts are away from the axis of symmetry.

In the context of using procedures students have developed previously for writing equations for quadratic functions from the information given in a table or a graph, students will examine the nature of the roots of quadratic functions and surface the need for non-real roots when the quadratic function does not intersect the x-axis. This task follows the approach of the historical development of these non-real numbers. As mathematicians developed formulas for solving quadratic and cubic polynomials, the square root of a negative number would sometimes occur in their work. Although such expressions seemed problematic and undefined, when mathematicians persisted in working with these expressions using the same algebraic rules that applied to realvalued radical expressions, the work would lead to correct results. In this task, students will be able to write the equation of quadratic #4 in both vertex and standard form, but attempting to use the quadratic formula to find the roots, and therefore the factored form, will produce expressions that contain the square root of a negative number. However, if students persist in expanding out this factored form using the usual rules of arithmetic, the non-real-valued radical expressions will go away, leaving the same standard form as that obtained by expanding the vertex form. This should give some validity to these non-real-valued radical expressions. It is suggested that these numbers not be referred to as “imaginary” numbers in this task, but only that they are noted to be problematic in the sense of not representing a real value.

The purpose of this task is to compare quadratic and exponential functions by examining tables and graphs for each. They will consider rates of change for each function type in various intervals and ultimately, see that an increasing exponential function will exceed a quadratic function.

The purpose of this task is to extend student understanding of the transformation of quadratic functions to include combinations of vertical stretches, reflections over the x-axis, and vertical and horizontal shifts. Students will write equations given story contexts, graphs, and tables. They will use their knowledge of transformations to graph equations and then they will apply their understanding to a general formula for the graph of a quadratic function in vertex form.

The purpose of this task is to develop understanding of the effect on the graph of a quadratic function of replacing !(!) by !(!) + !, !"(!), !(!") and !(! + !). The task begins with a brief story context to anchor student thinking about the effect of changing parameters on the graph. Students use technology to investigate the graphs, create tables and generalize about the transformations of quadratic functions.

Students will practice reflection/transltion of shapes with a card set and white boards. This is a lesson plan used to teach high school geometry. Materials are provided with the lesson.

This lesson provides an introduction to geometric transformations -- reflections, rotations, translations, and glide reflections. The accompanying applet allows students to perform a transformation and then analyze the relationship between the original object and the resulting image. A discussion of various kinds of symmetry is included. A student worksheet, lesson extensions, and guided discussions are provided.

One purpose of this task is to continue to solidify the definition of dilation: A dilation is a transformation of the plane, such that if O is the center of the dilation and a non-zero number k is the scale factor, then P’ is the image of point P if O, P and P’ are collinear. A second purpose of this task is to examine proportionality relationships between sides of similar figures by identifying and writing proportionality statements based on corresponding sides of the similar figures. A third purpose is to examine a similarity theorem that can be proved using dilation: a line parallel to one side of a triangle divides the other two proportionally.

One purpose of this task is to continue to solidify the definition of dilation: A dilation is a transformation of the plane, such that if O is the center of the dilation and a non-zero number k is the scale factor, then P’ is the image of point P if O, P and P’ are collinear.

A teacher's guide for teaching how to find the angles or lengths of the sides of a triangle, in addition to calculating the area of a triangle when some, but not all of these quantities are known. The document also provides practice exercises and an answer key.