Students explore quadratic functions by using a motion detector known as a Calculator Based Ranger (CBR) to examine the heights of the different bounces of a ball. Students will represent each bounce with a quadratic function of the form y = a (x - h)^2 + k. This lesson plan is based on the activity Tremain Nelson uses in the video for Part II of this workshop.

The purpose of this task is to develop and extend the concept of inverse functions in a linear context. In the task, students use tables, graphs, and equations to represent inverse functions as two different ways of modeling the same situation. The representations expose the idea that the domain of the function is the range of the inverse (and vice versa) for suitably restricted domains. Students may also notice that the graphs of the inverse functions are reflections over the ! = $ line, but with the understanding that the axes are part of the reflection.

In this lesson, students explore relationships between x-intercepts, factors, and roots of polynomial functions using the graphing calculator. Students also investigate rational functions, identifying the roots and the asymptotes as well as holes in the graphs. Students construct boxes of various dimensions using graph paper, collect height and volume data, and create a scatterplot in order to determine the height of the box with the maximum volume. Students can solve this problem using a graphing calculator or by using their own scatterplots drawn by hand. The use of questioning by the teacher, and the group work of the students are important features of this lesson.

The purpose of this task is to extend the Fundamental Theorem of Algebra from quadratic functions to cubic functions. The task asks students to use graphs and equations to find roots and factors and to consider the relationship between them. Students will also consider quadratic and cubic functions with multiple real roots and imaginary roots.

In this lesson, students will analyze and explore the data collected in the cell phone experiment. Graphs such as boxplots and comparative boxplots are drawn to illustrate the data. Measures of center (median, mean) and spread (range, Interquartile Range (IQR)) are computed. Outlier checks are performed. The distinction between independent samples and paired (matched) samples is discussed. Conclusions are drawn based upon the data analysis in the context of question(s) asked. An extension to a randomization test (permutation test) is discussed.

The purpose of this task is to use student understanding of log graphs and log expressions to derive properties of logarithms. In the task students are asked to find equivalent equations for graphs and then to generalize the patterns to establish the product, quotient, and power rules for logarithms.

The purpose of this task is for students to practice using the equation of the circle in different ways. In each case, they must draw inferences from the information given and use the information to find the equation of the circle or to justify conclusions about the circle. They will use the distance formula to find the measure of the radius and the midpoint formula to find the center of a circle.

In this task students consider the similarity of circles by examining two different transformation strategies that map one circle onto another. In the first strategy one circle is translated so that the center of the circles coincide. The inner circle can then be enlarged to carry it onto the outer circle, or the outer circle can be shrunk to carry it onto the inner circle. Students are asked to determine the scale factors for both the enlargement and the reduction. In the second strategy students observe that any circle can be mapped onto any other circle by dilation. Students are also asked to find the scale factor of this dilation, which is the same scale factor as the enlargement (or reduction) factor used in the first strategy. Students are also given the opportunity to draw similar figures inscribed within the two circles, which has the potential of surfacing some observations about central and inscribed angles, and the relationship between tangent lines and radii.

This resource has a set of problems for students to test their knowledge of calculating arc length, area of a sector and angle measures of a circle.

This purpose of this task is for students to connect their geometric understanding of circles as the set of all points equidistant from a center to the equation of a circle. In the task, students construct a circle using right triangles with a radius of 6 inches. This construction is intended to focus students on the Pythagorean Theorem and to use it to generate the equation of a circle centered at the origin. After constructing a circle at the origin, students are asked to use their knowledge of translations to consider how the equation would change if the center of the circle is translated.

The purpose of this task is to review and practice theorems and formulas associated with circles. Students will also draw upon ideas of similarity as well as right triangle trigonometry relationships to find the lengths of line segments. 30°-60°-90° triangles appear frequently in this task, so there is an opportunity to emphasize the relationships between the sides and how one can find the exact values of the lengths of the sides in this special right triangle.

Students will study transformations and the role transformations play in defining congruence. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.

Students build staircases out of square blocks. They look for patterns and come up with a rule or formula to predict the number of blocks in any staircase.

Students will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects, pictorial representations, and the properties of real numbers, equality, and inequality.

Students find a rule that agrees with a giventable. 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.

This lesson utilizes the concepts of cross-sections of three-dimensional models to demonstrate the derivation of two-dimensional shapes.

Students learn about basic recursion by exploring patterns in the data they generate from two simple probability-based experiments.

In the context of constructing circumscribed and inscribed circles for a triangle, students make observations about the relationships between central angles, inscribed angles and circumscribed angles. These observations are used to prove that angles inscribed in a semicircle are right angles, opposite angles of inscribed quadrilaterals are supplementary, and a tangent line to a circle is perpendicular to the radius drawn to the point of tangency.

In this lesson, students understand that when one group is randomly divided into two groups, the two groups’ means differ just by chance (a consequence of the random division). Students understand that when one group is randomly divided into two groups, the distribution of the difference in the two groups’ means can be described in terms of shape, center, and spread.