Chapter 2 – Graphing and Writing Linear Equations
2.1 Graphing Linear Equations
2.2 Slope of a Line
2.2b Slopes of Parallel and Perpendicular Lines
2.3 Graphing Linear Equations in Slope-Intercept Form
2.4 Graphing Linear Equations in Standard Form
2.5 Writing Equations in Slope-Intercept Form
2.6 Writing Equations in Point-Slope Form
2.6b Writing Equations of Parallel and Perpendicular Lines
2.7 Solving Real-Life Problems
8th Grade Curriculum
In Chapters 1 and 2, students extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students will understand the connections between proportional relationships, lines, and linear equations in these chapters. Also, students learn to apply the skills they acquired in Grades 6 and 7 with respect to symbolic notation and properties of equality to transcribe and solve equations in one variable and then in two variables.
Students begin by transcribing written statements using symbolic notation. Then, students write linear expressions leading to linear equations, which are solved using properties of equality. Students learn that not every linear equation has a solution. In doing so, students learn how to transform given equations into simpler ones until an equivalent equation results in a unique solution, no solution, or infinitely many solutions. Throughout the chapters, students must write and solve linear equations in real-world and mathematical situations
Students work with constant speed, a concept learned in Grade 6, but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane. It is here that students begin to investigate the shape of a graph of a linear equation. Students predict that the graph of a linear equation is a line and select points on and off the line to verify their claim. Also in these chapters is the standard form of a linear equation, ax + by = c, and when a and b do not equal zero, a non-vertical line is formed. Further, when a or b is equal to zero, then a vertical or horizontal line is produced.
Students come to know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph. In general, students learn that slope can be determined using any two distinct points on a line by relying on their understanding of properties of similar triangles. Students verify this fact by checking the slope using several pairs of points and comparing their answers. In this topic, students derive y = mx and y = mx + b for linear equations by examining similar triangles. Students generate graphs of linear equations in two variables first by completing a table of solutions and then by using information about slope and y-intercept. Once students are sure that every linear equation graphs as a line and that every line is the graph of a linear equation, students graph equations using information about x- and y-intercepts. Next, students learn some basic facts about lines and equations, such as why two lines with the same slope and a common point are the same line, how to write equations of lines given slope and a point, and how to write an equation given two points. With the concepts of slope and lines firmly in place, students compare two different proportional relationships represented by graphs, tables, equations, and descriptions. Finally, students learn that multiple forms of an equation can define the same line. Good Night!
7th Grade Curriculum
In Grade 6, students formed a conceptual understanding of integers through the use of the number line, absolute value, and opposites and extended their understanding to include the ordering and comparing of rational numbers (6.NS.C.5, 6.NS.C.6, 6.NS.C.7). In Chapters 1 and 2 I introduce the Integer Game: a card game that creates a conceptual understanding of integer operations and serves as a powerful mental model students can relay on during these chapters. Students build on their understanding of rational numbers to add, subtract, multiply, and divide signed numbers. Previous work in computing the sums, differences, products, and quotients of fractions and decimals serves as a significant foundation as well.
Students begin by returning to the number line to model the addition and subtraction of integers. They use the number line and the Integer Game to demonstrate that an integer added to its opposite equals zero, representing the additive inverse. Their findings are formalized as students develop rules for adding and subtracting integers, and they recognize that subtracting a number is the same as adding its’ opposite. Real-life situations are represented by the sums and differences of signed numbers. Students extend integer rules to include the rational numbers and use properties of operations to perform rational number calculations without the use of a calculator.
Students develop the rules for multiplying and dividing signed numbers. They use the properties of operations and their previous understanding of multiplication as repeated addition to represent the multiplication of a negative number as repeated subtraction. Students make analogies to the Integer Game to understand that the product of two negative numbers is a positive number. From earlier grades, they recognize division as the inverse process of multiplication. Thus, signed number rules for division are consistent with those for multiplication, provided a divisor is not zero. Students represent the division of two integers as a fraction, extending product and quotient rules to all rational numbers. They realize that any rational number in fractional form can be represented as a decimal that either terminates in 0’s or repeats. Students recognize that the context of a situation often determines the most appropriate form of a rational number, and they use long division, place value, and equivalent fractions to fluently convert between these fractions and decimal forms. We conclude by multiplying and dividing rational numbers using the properties of operations.View More