Curriculum 

Students use the properties of operations to add, subtract, multiply, and divide rational numbers. Then they operate with large positive numbers by writing repeated factors of 10, which creates a need for the properties and definitions of exponents. Students apply the properties of exponents to more efficiently find sums, differences, products, and quotients of large and small positive numbers written in scientific notation. They reason intuitively about square roots and cube roots and explore the Pythagorean theorem, which leads to writing numbers with square root notation, approximating roots, and defining irrational numbers. 

Students explore unknown angle measurement contexts that lead to equations and use if–then moves to solve equations in the forms p⁢x+q=r and 

p⁢(x+q)=r, where p, q, and r are specific rational numbers. They also multiply and factor algebraic expressions with rational numbers and apply and extend the if-then moves to solve inequalities. Students discover that linear equations in one variable can have only one solution, infinitely many solutions, or no solution, and they formalize the if–then moves as the properties of equality. Students apply multiplicative relationships and ratio reasoning to understand proportional relationships and identify them in tables, graphs, equations, and written descriptions. They compare proportional relationships and determine when constant rates indicate proportional relationships. Students then extend their understanding of proportional relationships to percents, recognizing a percent as a rate per 100. They use equations in the form ab=cd and y=k⁢x to solve percent problems in real-world contexts. 

Students construct triangles given certain conditions and determine whether triangles are identical and what conditions guarantee a unique triangle. They explore the proportional relationship between the circumference of a circle and its diameter and generalize the formulas for the circumference of a circle and the area of a circle. Students then experience rigid motions by using a transparency to represent the movement of the plane under a translation, reflection, or rotation. They define one figure as congruent to another if there is a sequence of rigid motions that maps the figure onto the other. Students apply rigid motions and the definition of congruent figures to establish facts about the angles created by parallel lines cut by a transversal, to determine the sum of the interior angle measures of a triangle, and to find the relationship between an exterior angle measure and the pair of remote interior angle measures of a triangle. They solve problems involving scale drawings of geometric figures. Students then analyze dilations and draw images of figures under dilations by using a variety of tools, understanding that similar figures are figures that can be mapped onto one another by using a sequence of rigid motions or dilations, or both. They develop the angle–angle criterion for similarity and solve for unknown side lengths of similar triangles in a variety of mathematical and real-world problems. 

Students recognize that there are an infinite number of solutions to a linear equation in two variables, and if they graph these solutions in a coordinate plane, the points form a line. Students use proportional relationships and similar triangles to develop an understanding of the slope of a line and then develop the slope-intercept form and the point-slope form of a linear equation. Then students transition to graphing systems of linear equations in two variables. They estimate the coordinates of the intersection point on the graph and verify that the ordered pair is a solution to the system. They also analyze systems of linear equations to determine the number of solutions. Needing a strategy for solutions composed of one or more fractional values, students use the substitution method to write a system of linear equations in two variables as one linear equation in one variable. Now equipped with various solution methods, students are challenged to write and solve systems resulting from numerical, geometrical, and real-world contexts. 

Students learn that a function relates inputs and outputs in such a way that each input is assigned one and only one output. They write equations to represent linear functions, and they relate the rate of change and initial value of each function back to the context. Students then transition to working with three-dimensional solids and use a variety of strategies to calculate the surface area of three-dimensional solids. They explore the cross sections of three-dimensional solids and use the information to understand how to compose or decompose a three-dimensional solid to calculate its volume more efficiently. Then, they develop the volume formulas for pyramids, cylinders, cones, and spheres and use linear functions to solve real-world problems involving volume. 

Students find empirical probabilities and compute theoretical probabilities. They estimate probabilities and observe that the more trials they conduct, the closer an empirical probability should be to the theoretical probability. They estimate a population proportion by using categorical data from a random sample and then compare populations with similar variability. Students use scatter plots to display bivariate numerical data. For data that appear to have a linear pattern, they draw a line that fits the data and write an equation of that line. Students examine bivariate categorical data by using two-way tables and find row or column relative frequencies to informally assess evidence of an association between two categorical variables.