In the context of examining combinations of cats and dogs that satisfy all of the “Pet Sitter” constraints students will solidify the following mathematics:
• The solution set to a system of inequalities represents the set of points that satisfy all of the inequalities simultaneously. That is, a point in the solution set of a system of inequalities makes all of the inequalities true. Contexts, such as “Pet Sitters” may imply additional constraints, such as y ≥ 0 or x ≥ 0, that are not explicitly stated.
• A shaded region on the coordinate grid is used to represent the solution set for a system of inequalities in two-variables. In this case, the points on the boundary lines that trace out the region are also included in the solution set, indicated by drawing a solid line for the boundaries. A shaded polygonal region is used to represent the solution to a system of constraints.

This is a visual demonstration of the sector-proof of the formula for the area of circle.

This lesson teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil).

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.

In this lesson, students calculate the rate of change between two points on a curve by determining the slope of the line joining those two points. They then extend this method to estimate the instantaneous rate of change at a given point by taking two points very close to and on either side of the given point. Although higher-level functions are used to illustrate situations, students are only calculating rate of change at specific intervals in this lesson.

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.

Students will research types of bridges, identify symmetry and types of functions and use linear functions to design a bridge.

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.

In this task students will develop insights into how to extend the process of solving equations—which they have previously examined for one- or two-step equations—so that the process works with multistep equations. They will observe that the process of solving an equation consists of writing a sequence of equivalent equations until the value(s) that will make each of the equations in the sequence true becomes evident. Each equation in the sequence of equivalent equations is obtained by operating on the expressions on each side of the previous equation in the same way, such as multiplying both sides of the equation by the same amount, or adding the same amount to both sides of the equation. This property of equality is often referred to as “keeping the equation in balance.” Our goal in each step of the equation solving process is to make the next equivalent equation contain fewer operations than the previous one by “un-doing” one operation at a time. When there are multiple operations involved in an equation, the order in which to “un-do” the operations can be somewhat problematic. This task examines ways to determine the sequence of “un-do-it” steps by using the structure of the equation.

This task solidifies the strategies for solving systems of equations that surfaced during the previous task. Students will begin by writing a system of equations to represent the shopping scenarios. Students will recognize that we can obtain an equivalent system of equations by replacing one or both equations in the system using one of the following steps:
• Replace an equation in the system with a constant multiple of that equation
• Replace an equation in the system with the sum or difference of the two equations
• Replace an equation with the sum of that equation and a multiple of the other

The goal of these steps is to obtain a system of equations in which the coefficient of one of the variables is the same in both equations. Then, when we subtract one of the equations from the other, we will obtain an equation that contains only one variable. This equation can be solved for its variable and the result can be substituted back into one of the original equations to obtain an equation that can be solved for the other variable.

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 focus of this task is on the generation of multiple expressions that connect with the visuals provided for the checkerboard borders. These expressions will also provide opportunity to discuss equivalent expressions and review the skills students have previously learned about simplifying expressions and using variables.

The purpose of this task is to solidify and extend the idea that geometric sequences have
a constant ratio between consecutive terms to include sequences that are decreasing (0 < r < 1).
The common ratio in one geometric sequence is a whole number and in the other sequence it is a
percent. This task contains an opportunity to compare the growth of arithmetic and geometric
sequences. This task also provides practice in writing and using formulas for arithmetic sequences.

This pacing guide is the collaborative work of math teachers, coaches, and curriculum leaders from 38 NC public school districts. The teams worked through two face-to-face meetings and digitally to compile the information presented. NC Math 1, 2, and 3 standards were used to draft possible units of study for these courses. This is a first draft living document. Teams plan to meet throughout the year to continually tweak, update and refine these guides. Updates will be posted as available to this google document.

This task builds upon students’ experiences with arithmetic and geometric sequences to
extend to the broader class of linear and exponential functions with continuous domains. The term
“domain” should be introduced and used throughout the whole group discussion. Students are
given contextual situations that can be modeled with either discrete and continuous linear
functions, or discrete and continuous exponential functions. They are also asked to compare these
types of functions using various representations.

The purpose of this task is to develop an understanding of the correlation coefficient. The task asks students to plot various data sets and use technology to calculate the correlation coefficient. They will order the graphs and create new data sets to develop the idea that the correlation coefficient indicates the strength and direction of a linear relationship in the data. Students also consider situations in which two variables are highly correlated, but the relationship is not necessarily causal.