# Math Curriculum – Grade 8

The middle school Math program is closely aligned with the New York State Learning Standards for Mathematics and share the same goals to provide students with the knowledge and understanding of mathematics necessary to function in a world very dependent upon the application of mathematics. Focus in the curriculum is meant to give students an opportunity to understand concepts and practice with them in order to reach a deep and fluent understanding. Coherence in the curriculum means progressions that span grade levels to build students’ understanding of ever more sophisticated mathematical concepts and applications. Rigor means a combination of fluency exercises, chains of reasoning, abstract activities, and contextual activities throughout the module.

The Mathematics standards presented by the New York State Learning Standards describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.

The New York State Learning Standards include six instructional shifts to facilitate student proficiency in mathematics with a focus on shifting instructional methodology on the Practice Standards.  The Practice standards include:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

For additional information regarding state standards go to the NYS Learning Standards for Mathematics.

In Grade 8, instructional time will focus on three critical areas:

1. Through their learning in the Number System, the Expressions, Equations, and Inequalities, and the Probability and Statistics domains, students:

• recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin;
• understand that the slope (m) of a line is a constant rate of change, as well as how the input and output change as a result of the constant rate of change;
• interpret a model in the context of the data by expressing a linear relationship between the two quantities in question and interpret components of the relationship (such as slope and y-intercept) in terms of the situation;
• solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line; and
• use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to represent, analyze, and solve a variety of problems.

2. Through their learning in the Functions and the Expressions, Equations, and Inequalities domains, students:

• grasp the concept of a function as a rule that assigns to each input exactly one output;
• understand that functions describe situations where one quantity determines another; and
• translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations of the function), and describe how aspects of the function are reflected in the different representations.

3. Through their learning in the Geometry domain, students:

• use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems;
• show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines;
• understand the statement of the Pythagorean Theorem and its converse, and why the Pythagorean Theorem holds; and
• apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons

Grade 8 Accelerated Mathematics/ Algebra I

Algebra I is the first mathematics course in high school and the focal point is functions; specifically linear, quadratic, and exponential functions. Students selected for Accelerated Mathematics participate in High School Algebra I. In Algebra I, students analyze and explain precisely the process of solving an equation. Students, through reasoning, develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations. Students learn the terminology specific to polynomials and understand that polynomials form a system analogous to that of integers. Students learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions represented graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on their understanding of integer exponents to consider exponential functions with integer domains. They compare and contrast linear and exponential functions, looking for structure in each and distinguishing between additive and multiplicative change. Students explore systems of linear and quadratic equations and linear inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions focusing in on the explicit forms of sequences written in subscript notation. In building models of relationships between two quantities, students analyze the key features of a graph or table of a function. Students strengthen their ability to discern structure in polynomial expressions. They create and solve equations involving quadratic and cubic expressions. Students reason abstractly and quantitatively in interpreting parts of an expression that represent a quantity in terms of its context; they also learn to make sense of problems and persevere in solving them by choosing or producing equivalent forms of an expression. Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They learn through repeated reasoning to anticipate the graph of a quadratic function by interpreting the structure of various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function, which may require students to write solutions in simplest radical form. Students expand their experience with functions to include more specialized functions—linear, exponential, quadratic, square, and those that are piecewise-defined, including absolute value and step. Students select from among these functions to model phenomena using the modeling cycle. Students build upon prior experiences with data, and are introduced to working with more formal means of assessing how a model fits data. Students display and interpret graphical representations of data, and if appropriate, choose regression techniques when building a model that approximates a linear relationship between quantities. They analyze their knowledge of the context of a situation to justify their choice of a linear model, compute and interpret the correlation coefficient, and distinguish between situations of correlation and causation.