Students develop facility with graphical interpretations of systems of equations and the …

Students develop facility with graphical interpretations of systems of equations and the meaning of their solutions on those graphs. For example, they can use the distance formula to find the distance between the centers of two circles and thereby determine whether the circles intersect in 0, 1, or 2 points. By completing the squares, students can convert the equation of a circle in general form to the center-radius form and, thus, find the radius and center. They can also convert the center-radius form to the general form by removing parentheses and combining like terms. Students understand how to solve and graph a system consisting of two quadratic equations in two variables.

Students understand the possibility that an equation—or a system of equations—has no …

Students understand the possibility that an equation—or a system of equations—has no real solutions. Students identify these situations and make the appropriate geometric connections.

Students solve quadratic equations with real coefficients that have complex solutions. They …

Students solve quadratic equations with real coefficients that have complex solutions. They recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

Students solve quadratic equations with real coefficients that have complex solutions. Students …

Students solve quadratic equations with real coefficients that have complex solutions. Students extend polynomial identities to the complex numbers. Students note the difference between solutions to the equation and the x-intercepts of the graph of said equation.

Students understand the Fundamental Theorem of Algebra; that all polynomial expressions factor …

Students understand the Fundamental Theorem of Algebra; that all polynomial expressions factor into linear terms in the realm of complex numbers. Consequences, in particular, for quadratic and cubic equations are understood.

Students will define sine and cosine as functions for all real numbers …

Students will define sine and cosine as functions for all real numbers measured in degrees. Students will evaluate the sine and cosine functions at multiples of 30 and 45.

Using tiles to represent variables and constants, learn how to represent and …

Using tiles to represent variables and constants, learn how to represent and solve algebra problem. Solve equations, substitute in variable expressions, and expand and factor. Flip tiles, remove zero pairs, copy and arrange, and make your way toward a better understanding of algebra.

The purpose of this task is to allow students to use their …

The purpose of this task is to allow students to use their prior understandings of similar triangles to develop an understanding of trigonometric ratios. As the discussion about the trigonometric ratios and their usefulness surfaces, be sure to name the ratios with their names (sine, cosine, tangent) and demonstrate how they are most frequently written as cos, sin, tan.

The purpose of this task is for students to connect the completing …

The purpose of this task is for students to connect the completing the square procedure learned in the previous two tasks to graphing parabolas. The task asks students to complete the square to change the equation of a quadratic function in standard form into vertex form. Students will need to extend their understanding of completing the square in an expression to an equation to maintain the equality, requiring that they add and subtract an equivalent term to one side of the equation (effectively adding zero) or that they add the same thing to both sides of the equation. After getting the equation into vertex form, students graph the equation of the parabola.

The purpose of this task is for students to solidify their understanding …

The purpose of this task is for students to solidify their understanding about inverse functions. Students will understand how to find the inverse of a function and know when to restrict the domain so that they can produce an invertible function from a non-invertible function. Students will also become familiar with square root functions as a result of this task and will connect square root functions to their domain, range, and graphs.

Susie is organizing the printing of tickets for a show. She has …

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

The purpose of this task is for students to solidify their understanding …

The purpose of this task is for students to solidify their understanding of piecewise functions using their background knowledge of domain and linear functions. Students will solidify their understanding of piecewise functions by

• answering questions relating to a piecewise function • using their knowledge of domain to graph each section of the piecewise function • graphing complete piecewise-defined functions from equations • interpreting the context of a piecewise function

Students explore quadratic functions by using a motion detector known as a …

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.

In this lesson, students explore relationships between x-intercepts, factors, and roots of …

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 develop students’ understanding of the …

The purpose of this task is to develop students’ understanding of the procedure of completing the square using area models. In the task, students will use diagrams of area models to make sense of the terms in a perfect square trinomial and discover the relationship between coefficients of a quadratic expression. They will use this understanding to complete the square to find equivalent forms of quadratic expressions. In this task, the quadratic expressions will be limited to those in which the coefficient of the !! term is 1. This task is the beginning of a learning cycle that ends in students using the completing the square procedure to find the vertex form of a quadratic function and graph the associated parabola.

Students learn a more formal definition of conditional probability and are asked …

Students learn a more formal definition of conditional probability and are asked to interpret conditional probabilities. Data are presented in two-way frequency tables, and conditional probabilities are calculated using column or row summaries.

In Math 8 students came to understand that a two-dimensional figure is …

In Math 8 students came to understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. In this task students are given two congruent figures and asked to describe a sequence of transformations that exhibits the congruence between them. While exploring potential sequences of transformations, students will notice how corresponding parts of congruent figures have to be carried onto one another, and they may look for ways that this can be accomplished in a consistent sequence of steps.

In this task students are asked to develop a strategy for locating …

In this task students are asked to develop a strategy for locating the center of a rotation, which leads to the observation that the center of rotation lies on the perpendicular bisectors of the segments joining image/pre-image pair of points. Such segments are defined to be chords of the circles on which the image/pre-image points are located. This work provides a method for finding the center of a circle. Students should also be able to prove that the perpendicular bisector of a chord contains a diameter of the circle and that the central angle formed by the radii that pass through the endpoints of the chord is bisected by the perpendicular diameter.

The purpose of this task is to practice thinking through and contributing …

The purpose of this task is to practice thinking through and contributing to the reasoning of others when working through a written geometric proof. Students will also practice all of the proof techniques discussed in this module: reasoning from a diagram; organizing a logical chain of reasoning from the given statements to the conclusion of the argument; and drawing upon postulates, definitions and previously proved statements to support the chain of reasoning. Three theorems are proved in this task: • The medians of a triangle meet at a point. • The angle bisectors of a triangle meet at a point. • The perpendicular bisectors of the sides of a triangle meet at a point.

It is not necessary for students to create these theorems from scratch, but they should be able to follow and extend the reasoning of the arguments presented, based on the diagrams and previously proved theorems. The ideas proved here are novel and somewhat surprising. These theorems should demonstrate the power of deductive reasoning to establish the validity of ideas that at first seem counterintuitive.

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