In the previous task we had to write multiple expressions to represent Carlos’ height on a Ferris wheel, depending upon which quadrant of the wheel he was located in. Our definition of sine needs to be modified if we want to use a single expression to represent Carlos’ height at every point on the wheel. The purpose of this task is to introduce a new definition of the sine function, based on an angle of rotation approach. This is the first extension of the definition of trigonometry—to define sine (and later, cosine and tangent) so the definitions apply to angles of rotation, and not just to the acute angles of right triangles.

The purpose of this task is to provide an opportunity for students to practice the ideas, strategies and representations that have surfaced and been examined in the previous tasks, 6.1-6.4. In the context of describing the periodic motion of the shadow of a rider on the Ferris wheel as the shadow moves back and forth across the ground when the sun is directly overhead, students will recognize that the cosine function can be used to measure the distance horizontally from the center of the wheel and they will derive the function horizontal position of the shadow = 25cos(18t).

In this lesson, students will collect their own data and visualize the distribution of the data by creating dotplots and generating comparative boxplots from the dotplot visualizations. They will also have an opportunity on comparing and contrasting visualizations to learn advantages and disadvantages of these two visualizations.

This lesson provides students with an introduction to exponential functions. The class first explores the world population since 1650. Students then conduct a simulation in which a population grows at a random yet predictable rate. Both situations are examples of exponential growth.

In this unit, students will review basic geometric vocabulary involving parallel lines, transversals, angles, and the tools used to create and verify the geometric relationships. They will also explore angle relationships formed when lines are cut by a transversal in city planning models

In four linked activities, students will apply their knowledge of ratios, proportions, fractions, decimals, percents, scientific notation, mean, median, mode, range, and pie graphs to interpret data and statistics regarding the U.S. government’s budget for prisons and correctional services. Then students will synthesize what they have learned and communicate it using diagrams and mathematical evidence.

In this task students use proportional reasoning to calculate arc length and the area of a sector relative to the circumference and area of the circle. The work of this task lays a foundation for the work of the next task in which proportionality relationships between arc length and radius are used to define radian measurement

Students will draw on their foundation of the analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students identify and make connections between zeros of polynomials and solutions of polynomial equations.

Students will continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

This online tutorial provides instruction and practice items on long division of polynomials.

The purpose of this task is to develop student ideas about solving exponential equations that require the use of logarithms and solving logarithmic equations. The task begins with students finding unknown values in tables and writing corresponding equations. In the second part of the task, students use graphs to find equation solutions. Finally, students build on their thinking with tables and graphs to solve equations algebraically. All of the logarithmic and exponential equations are in base 10 so that students can use technology to find values.

Susie is organizing the printing of tickets for a show. She has collected prices from several printers. Your task is to use graphs and algebra to advise Susie on how to choose the best printer.

Students carry out an interactive, geometric "proof without words" for the algebraic technique of completing the square in this interactive. The page also includes directions and a link to the final solution.

Students are provided with information and notice that the original three points given seem to be midpoints of the sides of a newly formed triangle. Their task is to determine how they would prove this conjecture? 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.

Students must work out how much you need to change the radius of a propane gas tank in order to double its capacity.

The purpose of this task is to solidify students’ understanding of the relationship between functions and their inverses and to formalize writing inverse functions. In the task, students are given a function and a particular value for input value #, and then asked to describe and write the function that that will produce an output that is the original # value. The task relies on students’ intuitive understanding of inverse operations such as subtraction “undoing” addition or square roots “undoing” squaring. There are two exponential problems where students can describe “undoing” an exponential function and the teacher can support the writing of the inverse function using logarithmic notation.

The purpose of this task is tie together what students have learned about roots, end behavior, operations, and graphs of polynomials. Each problem gives a few features of a polynomial and asks students to find other features, sometimes including the graphs. This will require them to know the end behavior of a polynomial, based on the degree, and to find all the roots, given some of the roots. In most cases, students are asked to write the equation of the polynomial or to write the equation in a different form than is given.